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IV054 CHAPTER 8: Elliptic Curves Cryptography and factorization
Cryptography based on manipulation of points of so called elliptic curves is getting
momentum and it has tendency to replace public key cryptography based on
unfeasibility to factorize integers or to compute discrete logarithm.
For example, US-government has recommended to use elliptic curve cryptography.
The main advantage of elliptic curves cryptography is that to achieve a certain level
of security shorter keys are required than in case of “classical cryptography”. Using
shorter keys can result in a considerable savings in hardware implementations.
The second advantage of elliptic curves cryptography is that quite a few of attacks
available for cryptography based on factorization and discrete logarithm do not
work for elliptic curves cryptography.
It is amazing how practical is elliptic curve cryptography that is based on very
strange theoretical concepts.
Elliptic curves cryptography
1
IV054 ElGamal cryptosystem
Design: choose a large prime p – (with at least 150 digits).
choose two random integers 1  q, x < p - where q is a primitive element of Z*p
calculate y = qx mod p.
Public key: p, q, y;
trapdoor: x
Encryption of a plaintext w: choose a random r and compute
a = qr mod p,
b = yr w mod p
Cryptotext: c = (a, b)
(Cryptotext contains indirectly r and the plaintext is masked by
multiplying with yr (and taking modulo p))
Decryption: w 
b
ax
mod p  ba  x mod p.
Proof of correctness: a x  q rx mod p
b
ax

yr w
ax

q rx w
q rx
 wmod p 
Note: Security of the ElGamal cryptosystem is based on infeasibility of the discrete
logarithm problem.
Elliptic curves cryptography
2
IV054 A group version of ElGamal cryptosystem


Given a group G,,   G,    i | i  0 .
k such that  k   , k  log  
ElGamal cryptosystem can be implemented in any group where discrete logarithm
problem is intractable.
Cryptosystem for (G, о),
Public key:  ,  .
Trapdoor: k such that  k  
Encryption: with a random r  1,...,  i | i  0 of a plaintext x:
 

ex, r    y1 , y2  where y1   r , y2  x   r
Decryption: of cryptotext (y1, y2):
1
d  y1 , y2   y2  y1k  x   r   rk  x   rk rk  x
An interesting fact is that discrete logarithm problem is intractable in any group Z*p,
where p is a prime, but it is easily computable in any group Z  p  , in spite of the fact
that for any p these two groups are isomorphic.
An important special case is that of the computation of discrete logarithm in a
group of points of an elliptic curve defined over a finite field.
 
Elliptic curves cryptography
3
IV054 Elliptic Curves
An elliptic curve E is the graph of the equation
E: y2 = x3 + ax + b
Where a, b will be, for our purposes, either rational numbers or integers (mod n)
extended by a “point at infinity”, denoted usually as ∞ (or 0) that can be regarded
as sitting, at the same time, at the top and bottom of y-axis.
We will consider mainly only those elliptic curves that have no multiple roots what is
equivalent the condition that 4a3+27b2 ≠ 0.
In case coefficients are rational numbers a graph of an elliptic curve has one of the
form shown in the following figure that depends on whether polynomial x3+ax+b
has three or one real root.
y2=x(x+1)(x-1)
Elliptic curves cryptography
y2=x3+73
4
IV054 Historical Remarks
Elliptic curves are not ellipses and therefore it seems strange that they have such
a name.
Elliptic curves actually received their names from their relation to so called elliptic
integrals
x2

x1
dx
x  ax  b
3
x2

x1
xdx
x 3  ax  b
that arise in the computation of the arc length of ellipses.
It may also seem puzzling why not to consider curves given by more general
equations
2
3
2
y  cxy  dy  x  ex  ax  b.
The reason is that if we are working with rational coefficients or mod p, where p>3
is a prime, then our general equation can be transformed to our special case. In
other cases it may be necessary to consider the most general form of equation.
Elliptic curves cryptography
5
IV054 ELIPTIC CURVES - GENERALITY
An elliptic curve over Z p mwhere p is a prime is the set of points (x,y)
satisfying Weierstrass equation
y  uxy  vx  x  ax  bx  c
2
3
2
for some constants u,v,a,b,c together with a single element O, called
the point of infinity.
If p≠2 Weierstrass equation can be simplified by transformation
y  y  (ux  v ) / 2
to get equation
y 2  x 3  dx 2  ex  f
for some constants d,e,f and if p≠3 by transformation
x  xd /3
to get equation
y  x  fx  g
2
Elliptic curves cryptography
3
6
IV054 Addition of Points on Elliptic Curves (1)
Geometry
On elliptic curves we can define addition of points in such a way that they form an
an Abelian group.
If the line through two different points P1 and P2 of an elliptic curve E intersects E
in a point Q=(x,y), then we define P1+P2=P3=(x,-y). (This also implies that for any
point P on E holds P+∞ = P.)
If the line through two different points P1 and P2 is parallel with y-axis, then we
define P1+P2=∞.
In case P1=P2, and the tangent to E in P1 intersects E in a point Q=(x,y), then we
define P1+P1=(x,-y).
It should now be obvious how to define subtraction of two points of an elliptic
curve.
It is now easy to verify that the above addition of points forms Abelian group with
∞ as the identity (null) element.
Elliptic curves cryptography
7
IV054 Addition of Points on Elliptic Curves (2)
Formulas
Addition of points P1=(x1,y1) and P2=(x2,y2) of an elliptic curve E: y2=x3+ax+b can
be easily computed using the following formulas:
P1 + P2 =P3=(x3,y3)
where
and
λ

x3 = λ2 - x1 – x2
y3 = λ(x1 – x3) – y1
( y 2  y1 ) /( x2  x1 )
( 3 x12  a ) /( 2 y1 )
If P1 ≠ P2
If P1 = P2
All that for the case that λ is finite; otherwise P3 = ∞.
Example For curve y2=x3+73 and P1=(2,9), P2=(3,10) we have P1 + P2 = P3= (-4,-3)
and P3 + P3 = (72,611).
Elliptic curves cryptography
8
IV054 Elliptic Curves mod n
The points on an elliptic curve
E: Y2=x3+ax+b (mod n)
are such pairs (x,y) mod n that satisfy the above equation, along with the point ∞
at infinity.
Example Elliptic curve y2=x3+2x+3 (mod 5) has points
(1,1),(1,4),(2,0),(3,1),(3,4),(4,0), ∞.
Example For elliptic curve E: y2=x3+x+6 (mod 11) and its point P=(2,7) holds
2P=(5,2); 3P=(8,3). Number of points on an elliptic curve (mod p) can be easily
estimated.
Hasse’s theorem If an elliptic curve E (mod p) has N points then |N-p-1|<2 p
The addition of points on an elliptic curve mod n is done by the same formulas as
given previously, except that instead of rational numbers c/d we deal with cd-1
Example For the curve E: y2=x3+2x+3 it holds (1,4)+(3,1)=(2,0); (1,4)+(2,0)=(?,?).
Elliptic curves cryptography
9
IV054 Elliptic Curves and Factorization
Let E be an elliptic curve and A, B its points such that B = kA = (A + A + … + A) for
some k. The task to find such a k is called discrete logarithm problem for the elliptic
curve E.
No efficient algorithm to compute discrete logarithm problem for elliptic curves is
known and also no good general attacks. Elliptic curves based cryptography is
based on these facts.
A general procedure for changing a discrete logarithm based cryptographic
protocol to a cryptographic protocol based on elliptic curves:
 Assign to the message (plaintext) a point on an elliptic curve.
 Change in the cryptographic protocol modular multiplication to addition of points
on an elliptic curve.
 Change in the cryptographic protocol exponentiation to multiplying a point on
elliptic curve by an integer.
 To the point of an elliptic curve that results from such a protocol assign a
message (cryptotext).
Elliptic curves cryptography
10
IV054 Mapping Messages into Points of Elliptic Curves (1)
Problem and basic idea
The problem of assigning messages to points on an elliptic curve is difficult
because there are no polynomial-time algorithms to write down points of an
arbitrary elliptic curve.
Fortunately there is a fast randomized algorithm to assign points of any elliptic
curve to messages that can fail with probability that can be made arbitrarily small.
Basic idea: Given an elliptic curve E (mod p), the problem is that not to every x
there is an y such that (x,y) is a point of E.
Given a message (number) m we therefore adjoin to m few bits at the end of m and
adjust them until we get a number x such that x3 + ax + b is a square mod p.
Elliptic curves cryptography
11
IV054 Mapping Messages into Points of Elliptic Curves ()
Technicalities
Let K be a large integer such that a failure rate of 1/2K is acceptable when trying to
encode a message by a point.
For j from 0 to K verify whether for x = mK + j, x3 + ax + b (mod p) is a square
(mod p) of an integer.
If such an j is found, encoding is done; if not the algorithm fails (with probability
1/2K because x3 + ax + b is a square approximately half of the time).
In order to recover the message m from the point (x,y), we compute:
x
 K 
Elliptic curves cryptography
12
IV054 Elliptic Curve Key Exchange
Elliptic curve version of the Diffie-Hellman key generation goes as follows:
Let Alice and Bob agree on a prime p, an elliptic curve E (mod p) and an point P on
E.
 Alice chooses an integer na, computes naP and sends it to Bob.
 Bob chooses an integer nb, computes nbP and sends it to Alice.
 Alice computes na(nbP) and Bob computes nb(naP). This way they have the same
key.
Elliptic curves cryptography
13
IV054 Elliptic Curve Version of ElGamal Cryptosystem
Standard version of ElGamal: Bob chooses a prime p, a generator q < p,
an integer a, computes y = qa (mod p), makes public p,q, y and keeps a secret.
To send a message m Alice chooses a random r, computes:
y1 = qr ; y2 = myr
and sends it to Bob who decrypts by calculating m  y2 y  a (mod p)
1
Elliptic curve version of ElGamal: Bob chooses a prime p, an elliptic curve
E (mod p), a point P on E, an integer a, computes Q = aP, makes E, p, and Q
public and keeps a secret.
To send a message m Alices expresses m as a point X on E, chooses random r,
computes
y1 = rP ; y2 = X + rQ
And sends the pair (y1,y2) to Bob who decrypts by calculating X = y2 – ay1.
Elliptic curves cryptography
14
IV054 Elliptic Curve Digital Signature
Eliptic curves version of ElGamal digital signatures has the following form under the
assumption that Alice wants to sign (a message) m, an integer, and to have signature verified
by Bob:
Alice chooses p and an elliptic curve E (mod p), a point P on E and calculates the number of
points n on E (mod p) – what can be done, and we assume that
0 < m < n. Alice then chooses a secret a and computes Q = aP. Alice makes public p, E, P, Q
and keeps secret a.
To sign m Alice does the following:
 Alice chooses a random integer r, 1 ≤ r < n such that gcd(r,n) = 1 and computes R = rP =
(x,y).
 Alice computes s = r–1(m – ax) (mod n)
 Alice sends the signed message (m,R,s) to Bob.
Bob verifies the signature as follows:
 Bob declares the signature as valid if xQ + sR = mP
The verification procedure works because
xQ + sR = xaP + r–1(m – ax)(rP) = xaP + (m – ax)P = mP
Warning Observe that actually rr–1 = 1 + tn for some t. For the above verification procedure to
work we then have to use the fact that nP = ∞ and P + t ∞ = P
Elliptic curves cryptography
15
IV054 Factoring with Elliptic Curves
Basis idea: To factorize an integer n choose an elliptic curve E, a point on E (mod
n) and compute either iP for i=2,3,4,… or 2j P for j=1,2,…. In doing that one needs
to compute gcd(k,n) for various k. If one if these values is between 1 and n we
have a factor of n.
Factoring of large integers: The above idea can be easily parallelised and
converted for using enormous number of computers to factor very large n. Each
computer gets some number of elliptic curves and some points on them and
multiplies these points by some integers according a given rule. If one of computes
encounters during such a computation a need to compute 1<gcd(k,n)<n
factorization is finished.
Example: If curve E: y2 = x3 + 4x + 4 (mod 2773) and its point P=(1,3) is used, then
2P=(1771,705) and in order to compute 3P one has to compute
gcd(1770,2773)=59 and factorization is done.
Example: For elliptic curve E: y2=x3+x+1 (mod 35) and its point P=(1,1) we have
2P=(2,2); 4P=(0,22); 8P=(16,19) and at the attempt to compute 9P one needs to
compute gcd(15,35)=15 and again the factorization is done. The only things that
remains to be explored is how efficient is this method and when it is more efficient
than other methods.
Elliptic curves cryptography
16
IV054 Important Observations (1)
 If n = pg for primes p,q, then an elliptic curve E (mod n) can be seen as a pair of
elliptic curves E (mod p) and E (mod q).
 It follows from the Lagrange theorem that for any elliptic curve E (mod n) and its
point P there is a k<n such that kP = ∞.
 In case of an elliptic curve E (mod p) for some prime p, the smallest positive
integer m such that mP = ∞ for some point P divides the number N of points on the
curve E (mod p). Hence NP = ∞.
 If N is a product of small primes, then b! will be a multiple of N for a reasonable
small b. Therefore, b!P = ∞.
 The number with only small factors is called smooth and if all factors are smaller
than an b, then in is called b-smooth.
 It can be shown that the density of smooth integers is so large that if we choose a
random elliptic curve E (mod n) then it is a reasonable chance that N is smooth.
Elliptic curves cryptography
17
IV054 Practicality of Factoring Using ECC (1)
Let us continue to discuss the following key problem for factorization using elliptic
curves:
Problem: How to choose k such that for a given point P we should try to compute
iP or 2iP for multiples of P smaller than kP?
Idea: If one searches for m-digits factors, one should choose a k in such a way that
k is a multiple of as many of m-digit numbers as possible which do not have too
large prime factors. In such a case one has a good chance that k is a multiple of
the number of elements of a group of points of an elliptic curve modulo n.
Method: One chooses an integer B and takes as k the product of all maximal
powers of primes smaller than B.
Example: In order to find a 6-digit factor one chooses B=147 and k=27∙34∙53 ∙
72∙11∙2∙13∙… ∙139. The following table shows B and the number of elliptic curves
one has to test:
Elliptic curves cryptography
18
IV054 Practicality of Factoring Using ECC (2)
Digits of to-be-factors
6
9
12
18
24
30
B
147
682
2462
23462
162730
945922
Number of curves
10
24
55
231
833
2594
Computation time by the elliptic curves method depends on the size of factors.
Elliptic curves cryptography
19
IV054 Elliptic Curves: FAQ
 How to choose an elliptic curve E and point P on E? An easy way is first choose
a point P(x,y) and an a and then compute b = y2 - x3 - ax to get curve
E: y2 = x3 + ax + b.
 What happens at the factorization using elliptic curve method if for a chosen
curve (E mod n) the corresponding cubic polynomial x3 + ax + b has multiple roots
(that is if 4a3 + 27b2 = 0) ? No problem, method still works.
 What kind of elliptic curves are really used in cryptography? Elliptic curves over
fields GF(2n) for n > 150. Dealing with such elliptic curves requires, however,
slightly different rules.
Elliptic curves cryptography
20
IV054 Factorization of Fermat numbers
Factorization of so-called Fermat numbers 22^i + 1 is a good example to illustrate
progress that has been made in the area of factorization.
Pierre de Fermat (1601-65) expected that all numbers
Fi = 2j + 1, j=2i
i1
are primes.
This is true for i = 1,…,4. F1 = 5, F2 = 17, F3 = 257, F4 = 65537.
1732 L. Euler found that F5 = 4294967297 = 641 · 6700417
1880 Landry+LeLasser found that
F6 = 18446744073709551617 = 274177 · 67280421310721
1970 Morrison+Brillhart found factorization for F7 =(39 digits)
F7 = 340282366920938463463374607431768211457 =
= 5704689200685129054721 · 59649589127497217
1980 Brent+Pollard found factorization for F8
1990 A. K. Lenstra+… found factorization for F9 (155 digits)
Fermat test: If x n1  1mod n , then n is not prime.
Elliptic curves cryptography
21
IV054 POLLARD’s p-1 algorithm
Pollard’s algorithm (to factor n given a bound b).
a := 2;
for j=2 to b do a:= aj mod n;
f:= gcd(a-1,n);
if 1 < f < n then f is a factor of n otherwise failure
Indeed, let p be a prime divisor of n and q < b for every prime q|(p-1).
(Hence (p-1)|b!).
At the end of the for-loop we therefore have
a Ξ 2b! (mod n)
and therefore
a Ξ 2b! ( mod p)
By Fermat theorem 2p-1 Ξ 1 (mod p) and since (p-1)|b! we have that p|(a-1)
and therefore
p|d = gcd(a-1,n)
Elliptic curves cryptography
22
IV054 Important Observations (2)
 The advantage of elliptic curve factorization method over the p-1 method is the
following.
The p-1 method requires that p-1 is smooth. The elliptic curve method requires only
that there are enough smooth integers near p and so at least one of randomly
chosen integers near p is smooth.
This means that the elliptic curves factorization method succeeds much more often
than p-1 method.
Elliptic curves cryptography
23
IV054 Method of quadratic sieve to factorize n
Basic idea: One finds x, y such that n | (x2 - y 2)
Reasoning: If n divides (x + y)(x - y) and n does not divide neither x+y nor x-y, then
one factor of n has to divide x+y and another one x-y.
Example
n = 7429 = 2272 -2102,
x – y = 17
gcd(17, 7429) = 17
x = 227, y = 210
x + y = 437
gcd(437, 7429) = 437.
How to find x and y? One forms a system of (modular) linear equations and
determines x and y from the solutions of such a system.
number of digits of n 50
60
70
80
90
100
110
120
number of equations 3000 4000 7400 15000 30000 51000 120000 245000
Elliptic curves cryptography
24
IV054 Method of quadratic sieve to factorize n
Step 1 One finds numbers x such that x2 - n is small and has small factors.
Example
832 – 7429 = -540 = (-1) · 22 · 33 · 5
872 – 7429 = 140 =
22 · 5 · 7
relations
882 – 7429 = 315 =
32 · 5 · 7
Step 2 One multiplies some of the relations if their product is a square.
For example
(872 – 7429)(882 – 7429) = 22 · 32 · 52 · 72 = 2102
Now
(87 · 88)2  (872 - 7429)(882 - 7429) mod 7429
2272  2102 mod 7429
Hence 7429 divides 2272-2102.
Formation of equations: For the i-th relation one takes a variable i and forms the expression
((-1) · 22 · 33 · 5)1 · (22 · 5 · 7)2 · (32 · 5 · 7)3 = (-1)1 · 221 + 22 · 321 + 22 · 51 + 2 + 3 · 72 +3
If this is to form a quadrat the
following equations have to hold
.
Elliptic curves cryptography
1
 0 mod 2
 1   2   3  0 mod 2
 2   3  0 mod 2
 1  0,  2   3  1
25
IV054 Method of quadratic sieve to factorize n
Problem How to find relations?
Using the algorithm called Quadratic sieve method.
Step 1 One chooses a set of primes that can be factors - a so-called factor basis.
One chooses an m such that m2 - n is small and considers numbers (m + u)2 - n for
–k  u  k for small k.
One then tries to factor all (m + u)2 - n with primes from the factor basis, from the
smallest to the largest.
u
(m + u)2 - n
Sieve with 2
Sieve with 3
Sieve with 5
Sieve with 7
-3
-3
-3
-540 -373 -204
-135
-51
-5
-17
-1
0
-33
-11
1
2
3
140 315 492
35
123
35 41
7
7
1
1
In order to factor a 129-digit number from the RSA challenge they used
8 424 486 relations
569 466 equations
544 939 elements in the factor base
Elliptic curves cryptography
26
IV054 The rho method of integer factorization
Basic idea 1. Choose an easy to compute f: Zn  Zn and x0 Zn.
Example f(x) = x2 + 1
2. Keep computing xj+1 = f(xj), j = 0,1,2,… and gcd(xj - xk, n), k j.
(Observe that if xj  xk mod r for a prime factor r of n, then gcd(xj - xk, n)  r.)
Example n = 91, f(x) = x2+1, x0 = 1, x1 = 2, x2 = 5, x3 = 26
gcd(x3 - x2, n) = gcd(26 - 5, 91) = 7
Remark: In the rho method it is important to choose f in such a way that f maps
Zn into Zn in a ”random'' way.
Basic question: How good is the rho method?
(How long we expect to have to wait before we get two values xj, xk such that
gcd(xj - xk, n)  1 if n is not a prime?)
Elliptic curves cryptography
27
IV054 Basic lemma
Given: n, f:Zn  Zn and x0Zn
We ask how many iterations are needed to get xj  xk mod r where r is a prime
factor of n.
Lemma Let S be a set, r = |S|. Given a map f:S  S, x0S, let xj+1 = f(xj), j  0. Let
 > 0, l  1   2 r . Then the proportion of pairs (f, x0) for which x0, x1,…, xl are
distinct, where f runs over all mappings from S to S and x0 over all S, is less than e-.
Proof Number of pairs (x0, f) is r r+1.
How many pairs (x0, f) are there for which x0,…, xl are distinct?
r choices for x0, r-1 for x1, r-2 for x2,…
The values of f for each of the remaining r - l values are arbitrary - there are r r - l
possibilities for those values.
Total number of ways of choosing x0 and f such that x0,…, xl are different is
r
r l
l
 r  j 
j 0
l 1
and the proportion of pairs with such a property is r  j 0 r  j .
l

 

For l  1  2lr  we have ln r l 1  j 0 r  j   ln  j 0 1  rj    j 1  rj   l 2l r1
l
l
l
  2l r   
2
Elliptic curves cryptography
2 r
2r
2  .
28
IV054 RHO-ALGORITHM
A simplification of the basic idea: For each k compute gcd(xk - xj, n) for just one j < k.
Choose f:Zn  Zn, x0, compute xk = f(xk-1), k > 0.
If k is an (h +1)-bit integer, i.e. 2h  k  2h+1, then compute gcd(xk, x2^h-1).
Example n = 4087, f(x) = x2 + x + 1, x0 = 2
x1 = f(2) = 7,
gcd(x1 - x0, n) = 1
x2 = f(7) = 57,
gcd(x2 - x1, n) = gcd(57 – 7, n) = 1
x3 = f(57) = 3307,
gcd(x3 - x1, n) = gcd(3307 - 7, n) = 1
x4 = f(3307) = 2745,
gcd(x4 - x3, n) = gcd(2745 - 3307, n) = 1
x5 = f(2746) = 1343,
gcd(x5 - x3, n) = gcd(1343 - 3307, n) = 1
x6 = f(1343) = 2626,
gcd(x6 - x3, n) = gcd(2626 - 3307, n) = 1
x7 = f(2626) = 3734,
gcd(x7 - x3, n) = gcd(3734 - 3307, n) = 61
Disadvantage We likely will not detect the first case such that for some k0 there is a
j0 < k0 such that gcd(xk0 - xj0, n) > 1.
This is no real problem! Let k0 has h +1 bits. Set j = 2h+1 -1, k = j + k 0 - j0. k has
(h+2) bits, gcd(xk - xj, n) > 1
k < 2h+2 = 4 · 2h  4k0.
Elliptic curves cryptography
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IV054 RHO-ALGORITHM
Theorem Let n be odd + composite and 1 < r < sqrt(n) its factor. If f, x0 are chosen
randomly, then rho algorithm reveals r in 4 n log 3 n  bit operations with high
probability. More precisely, there is a constant C > 0 such that for any  > 0, the
probability that the rho algorithm fails to find a nontrivial factor of n in C  4 n log 3 n
bit operations is less than e - .
Proof Let C1 be a constant such that gcd(y - z, n) can be computed in C1log3n bit
operations whenever y, z < n.
Let C2 be a constant such that f(x) mod n can be computed in C2log2n bit
operations if x < n.
If k0 is the first index for which there exists j0 < k0 with xk0  xj0 mod r, then the rhoalgorithm finds r in k 4k0 steps.
The total number of bit operations is bounded by -> 4k0(C1log3n + C2log2n)
By Lemma the probability that k0 is greater than 1  2 r is less than e - .
If k0  1  2 r , then the number of bits operations needed to find r is bounded by
4 1  2 r C1 log 3 n  C2 log 2 n   4 1  2 4 n C1 log 3 n  C2 log 2 n 




3
4
If we choose C > 4sqrt(2)(C1 + C2), then we have that r will be found in C  n log n
bit operations - unless we made uniformed choice of (f, x0) the probability of what is
at most e - .
Elliptic curves cryptography
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IV054 Simple factorization strategy to factor an integer n
1.For i = 3, 5,… till [10logn] check whether i |n.
If such an i is found we have a factor. Otherwise:
2. Fermat test:
Verify whether 2n-1  1 mod n.
If yes, n is probably prime. To confirm it use Lucas test.
3. Lucas test:
Lucas sequence: U0 = 0, U1 = 1, Ui + 1 = Ui – qUi - 1, i  1.
Lucas theorem: If n is prime, n>q, (1 - 4q|n) = -1, then n|Un+1.
Test: Find the smallest D such that (D|n) = -1, put D = 1 - 4q, check whether
Un+1  0 mod n. If not, n is composite. Otherwise n is prime with large
probability.
Remark No composite integer is known that would satisfy both Fermat and
Lucas tests. (A proof of this fact exists for n < 25 · 109.)
Homework: Factorize: 7500596246954111183.
Elliptic curves cryptography
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IV054 Computation of Un+1
V0  2
U 2t  U tVt
U 2t 1 
U 2t  V2t
2
U0  2
V2t  Vt 2  2 g t
V2t 1 
DU 2t  V2t
2
Homework
1. Factor 277 – 3
2. Factor 279 – 3
Elliptic curves cryptography
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IV054 Factorization of a 512-bit number
On August 22, 1999, a team of scientifists from 6 countries found, after 7
months of computing, using 300 very fast SGI and SUN workstations and
Pentium II, factors of the so-called RSA-155 number with 512 bits (about 155
digits).
RSA-155 was a number from a Challenge list issue by the US company RSA
Data Security and “represented'' 95% of 512-bit numbers used as the key to
protect electronic commerce and financinal transmissions on Internet.
Factorization of RSA-155 would require in total 37 years of computing time on
a single computer.
When in 1977 Rivest and his colleagues challenged the world to factor RSA129, he estimated that, using knowledge of that time, factorization of RSA-129
would require 1016 years.
Elliptic curves cryptography
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IV054 LARGE NUMBERS
Hindus named many large numbers - one having 153 digits.
Romans initially had no terms for numbers larger than 104.
Greeks had a popular belief that no number is larger than the total count of sand
grains needed to fill the universe.
Large numbers with special names:
googol - 10100
golplex - 1010^100
FACTORIZATION of very large NUMBERS
W. Keller factorized F23471 which has 107000 digits.
J. Harley factorized: 1010^1000 +1.
One factor: 316,912,650,057,350,374,175,801,344,000,001
1992 E. Crandal, Doenias proved, using a computer that F22, which has more than
million of digits, is composite (but no factor of F22 is known).
34
1010
Number 10
was used to develop a theory of the distribution of prime numbers.
Elliptic curves cryptography
34